Lecture Notes on Polynomials 📝

Introduction

• Instructor: Shobit Bhaiya
• Chapter: Class 9, Chapter 2: Polynomials
• Importance: Thorough understanding for solving NCERT, RD Sharma, RS Aggarwal, and other question banks.
• Goal: Aim for 100/100 in exams.

Preparation

• Supplies: Bring a copy and pen to take notes.
• Mindset: High focus and conceptual understanding needed.

Basic Terms

Variables

• Represent unknown values, e.g. x, y, z, p, q, r, s, t.
• Definition: A value that can change or is unknown.
  • Examples: Temperature (varies), unknown quantities.
  • Denotation: Small alphabetical letters like x, y, t.

Constants

• Values that are fixed, e.g. 2, π, -1.
• Examples: 2 is constant everywhere (India, Australia, USA, etc.).
• Definition: A fixed value that doesn't change.

Algebraic Expression

• Definition: Combination of terms using + and -.
  • Terms: A single variable, a constant, or a product of variables and constants.
  • Example: 2x - 3y + 4
    • Identifying Terms: 2x, -3y, +4 are each terms.
• Combining Terms: Terms combined using + and - are algebraic expressions.

Polynomials

• Definition: A special form of algebraic expressions where variables have whole number powers.
  • Whole Numbers: 0, 1, 2, 3, ...
  • Example: 4x - 3 where whole number powers means it's a polynomial.
• Non-examples: Expression with fractional powers (e.g., x^1/2) or negative powers (e.g., x^-1).

General Form of Polynomial

• Polynomial in a variable x represented as P(x) = ax^n + bx^(n-1) + ... + z.
• Notation: In the format P(x) or Q(y) depending on the variable.
  • Degree: Highest power of variable in the polynomial.

Identifying Polynomials

• Example Checking: Verify if variable powers are whole numbers to confirm polynomial identity.
• Degree Calculation: Highest power among the terms is the degree of the polynomial.

Types of Polynomials

Based on Number of Terms

1. Monomial: 1 term (e.g., 6x, 3yz)
2. Binomial: 2 terms (e.g., 5x + 4, 2a - 3b)
3. Trinomial: 3 terms (e.g., x^2 + 3x + 4)

Based on Degree

1. Linear Polynomial: Degree 1 (e.g., 2x + 1)
2. Quadratic Polynomial: Degree 2 (e.g., 3x^2 + 2x + 1)
3. Cubic Polynomial: Degree 3 (e.g., x^3 + 3x^2 + 2x + 1)

Constant & Zero Polynomials

• Constant Polynomials: Degree = 0, non-zero (
• Zero Polynomial: 0, Degree is not defined.

Value of a Polynomial

• Definition: Polynomial value at a specific point P(a), plug a in place of x in P(x).
• Example:
  • P(x) = 3x^2 + 4, find P(2): P(2) = 3*2^2 + 4 = 12 + 4 = 16.

Zeros of Polynomials

• Definition: Values of x for which P(x) = 0.
• Example: For 3x - 3, find zero: 3x - 3 = 0 => x = 1.

Long Division Method

• Process: For dividing polynomials, similar to numerical long division.
• Steps: Align degrees, multiply divisor to match leading term, subtract, repeat.
• Remainder Theorem: For P(x) divided by x - a, remainder is P(a).

Factor Theorem

• Definition: x - a is a factor if P(a) = 0.
• Examples: Checking for factors by testing zeros.

Polynomial Factorization

Splitting Middle Term Method

• Used for Quadratic Polynomials: Split the middle term into two terms such that their sum is the middle term and their product is the constant term.
• Example: x^2 - 7x + 12 => x^2 - 4x - 3x + 12 => (x-4)(x-3).

Using Factor Theorem

• Process: Identify zero of polynomial, factor out x - a, divide polynomial, repeat.
• Example: x^3 - 6x^2 + 11x - 6 using P(1) = 0.

Algebraic Identities

• Square of Sum: (a+b)^2 = a^2 + 2ab + b^2
• Square of Difference: (a-b)^2 = a^2 - 2ab + b^2
• Product of Sums and Differences: (a+b)(a-b) = a^2 - b^2
• Cube of Sum/Difference: (a+b)^3 = a^3 + b^3 + 3ab(a+b) and similar for difference.
• Summary: Various algebraic identities to simplify polynomials.

Practical Applications and Examples

• Example Problems: Detailed example problems using above identities and theorems.
• Exercises: Practice problems from textbooks; verify answers using learned methods.

Summary and Strategy

• Review Key Points: Focus on understanding concepts, practicing varied problems.
• Use Correct Terminology: Communicate solutions effectively and correctly.
• Consistent Practice: Regular practice to excel in exams.

Conclusion

• Encouragement: Practice makes perfect, build a solid foundation in polynomials.
• Next Steps: Practice problems, review notes, seek help if needed.
• Goodbye: Wishing success in mastering polynomials.